Would like to know if it's wrong to use "my something of choice"? Does the phrase have the same meaning as "something of my choice"?
The two phrases have slightly different connotations.
"My something of choice" implies a personal favorite or habitual preference. E.g. "The golden gun is his weapon of choice."
"Something of my choice" implies getting to choose from a specific set. It is appropriate for describing a momentary decision. E.g. "The steak comes with a side of your choice: soup of the day, Caesar salad, or steamed vegetables."
The phrase the something of choice means
In the same vein, my something of choice is what you prefer to use in a given situation, e.g., "The Porsche Panamera is my car of choice for long journeys."
On the other hand, something of my choice is something you do because you want to, not because you have to: "Bicycling thirty miles to work is of my choice."